Regular variation and differential equations: Theory and methods

Transcription

1 Regular variation and differential equations: Theory and methods Pavel Řehák Institute of Mathematics Czech Academy of Sciences External meeting IM CAS, Brno, November 2016 Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 1 / 40

2 Structure of the talk Regular variation Karamata theory De Haan theory Regular variation and differential equations Selected results Emden-Fowler type systems Half-linear differential equations Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 2 / 40

3 Theory of regularly varying functions initiated by J. Karamata (1930). But there are also earlier works... study of relations such that f (λt) lim = g(λ) (0, ), λ > 0, t f (t) together with their applications (integral transforms Tauberian theorems, probability theory, number theory, complex analysis, differential equations, etc.) N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27, Cambridge University Press, W. Feller, An Introduction to Probability Theory and Its Applications, vol. II, Wiley, NY, J. L. Geluk, L. de Haan, Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Amsterdam, E. Seneta, Regularly Varying Functions, Lecture Notes in Mathematics 508, Springer-Verlag, Berlin-Heidelberg-New York, Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 3 / 40

4 Definition A measurable function f : [a, ) (0, ) is called regularly varying (at ) of index ϑ if for all λ > 0, f (λt) lim t f (t) = λϑ. [Notation: f RV(ϑ)] If ϑ = 0, then f is called slowly varying. [Notation: f SV] [RV 0 means regular variation at zero.] The Uniform Convergence Theorem If f RV(ϑ), then the relation in the definition holds uniformly on each compact λ-set in (0, ). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 4 / 40

5 Representation Theorem f is regularly varying of index ϑ if and only if { } t δ(s) f (t) = ϕ(t) exp s ds where ϕ(t) const > 0 and δ(t) ϑ as t. f is regularly varying of index ϑ if and only if { } t f (t) = t ϑ ψ(s) ϕ(t) exp ds s where ϕ(t) const > 0 and ψ(t) 0 as t. a a If ϕ(t) const, then f is said to be normalized regularly varying (f N RV). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 5 / 40

6 Examples of (non-)sv functions f is regularly varying of index ϑ if and only if f (t) = t ϑ L(t), where L SV. n i=1 (ln i t) µ i, where lni t = ln ln i 1 t and µ i R is SV function. 2 + sin(ln 2 t) and (ln Γ(t))/t are SV functions. 1 t t a 1 ln s ds is SV function. SV functions may exhibit infinite oscillation { (i.e., lim inf t } L(t) = 0, lim sup t L(t) = ), for example, exp (ln t) 1 3 cos(ln t) sin t, 2 + sin(ln t) are NOT SV functions. exp t is NOT RV function. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 6 / 40

7 Extension in a logical and useful manner of the class of functions whose asymptotic behavior is that of a power function, to functions where asymptotic behavior is that of a power function multiplied by a factor which varies more slowly than a power function. RV generalized power laws; (asymptotic scale invariant property) f (λt) = g(λ)f (t) f (λt) g(λ)f (t) as t Regularly varying functions have a good behavior with respect to integration and many other pleasant properties. Regularly varying functions and related objects naturally occur in differential equations, and are very useful in the study of their qualitative properties. V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics 1726, Springer-Verlag, Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 7 / 40

8 Other selected properties If L 1, L 2 SV, then L α 1 SV (with α R), L 1L 2 SV, L 1 + L 2 SV and L 1 L 2 SV (provided L 2 (t) ). If L SV and ϑ > 0, then t ϑ L(t), t ϑ L(t) 0 as t. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 8 / 40

9 Other selected properties (Almost monotonicity) For a positive measurable function L it holds: L SV if and only if, for every ϑ > 0, there exist a (regularly varying) nondecreasing function F and a (regularly varying) nonincreasing function G with t ϑ L(t) F (t), t ϑ L(t) G(t) as t. (Asymptotic inversion) If g RV(ϑ) with ϑ > 0, then there exists g RV(1/ϑ) with f (g(t)) g(f (t)) t as t. Here g (an asymptotic inverse of f ) is determined uniquely to within asymptotic equivalence. One version of g is the generalized inverse f (t) := inf{s [a, ) : f (s) > t}. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 9 / 40

10 Other selected properties (Karamata s theorem) (Karamata s theorem; direct half) (i) If L SV and γ < 1, then t s γ L(s) ds (ii) If L SV and γ > 1, then t a 1 γ 1 t γ+1 L(t) as t. s γ L(s) ds 1 γ + 1 t γ+1 L(t) as t. (iii) The integral L(s)/s ds may or may not converge. The function a L(t) = t a L(s)/s ds or L(t) = L(s)/s ds is a new SV function and such t that L(t)/ L(t) 0 as t. (Karamata s theorem; converse half) Opposite directions in (i) and (ii). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 10 / 40

11 Rapid variation Definition A measurable function f : [a, ) (0, ) is called rapidly varying of index, we write f RPV( ), if { f (λt) lim t f (t) = 0 for 0 < λ < 1, for λ > 1, and is called rapidly varying of index, we write f RPV( ), if { f (λt) lim t f (t) = for 0 < λ < 1, 0 for λ > 1. The class of all rapidly varying solutions is denoted as RPV. While RV functions behaved like power functions (up to a factor which varies more slowly ), RPV functions have a behavior close to that of exponential functions. Regularly bounded functions, Zygmund class,..., RV in other settings,... Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 11 / 40

12 De Haan theory, class Π Definition A measurable function f [a, ) R is said to belong to the class Π if there exists a function w : (0, ) (0, ) such that for all λ > 0, f (λt) f (t) lim = ln λ; t w(t) we write f Π or f Π(w). The function w is called an auxiliary function for f. The class Π of functions f is, after taking absolute values, a proper subclass of SV. The de Haan theory is both a direct generalization of the Karamata theory and what is needed to fill certain gaps, or boundary cases, in Karamata s main theorem. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 12 / 40

13 Class Π selected properties If f Π, then for 0 < c < d < the relation in the definition holds uniformly for λ [c, d]. If f Π(L), then as t.... L(t) f (t) 1 t t a f (s) ds Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 13 / 40

14 De Haan theory, class Γ Definition A nondecreasing function f : R (0, ) is said to belong to the class Γ if there exists a function v : R (0, ) such that for all λ R f (t + λv(t)) lim = e λ ; t f (t) we write f Γ or f Γ(v). The function v is called an auxiliary function for f. A function is said to belong to the class Γ (v) if 1/f Γ(v). It holds: Γ ± RPV(± ) Inversion of Π, but there are also other way of understanding. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 14 / 40

15 Class Γ selected properties If f Γ, then the relation in the definition holds uniformly on compact λ-sets. f Γ if and only if If f Γ(v), then lim t s 0 0 f (t) t f (τ) dτ ds ( ) t 2 = 1. 0 f (s) ds v(t + λv(t)) v(t) as t locally uniformly in λ R (i.e., v SN ; self-neglecting).... Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 15 / 40

16 De Haan theory, Beurling slow variation Definition A measurable function f : R (0, ) is Beurling slowly varying if we write f BSV. Definition f (t + λf (t)) lim = 1 for all λ R; t f (t) If the relation holds locally uniformly in λ, then f is called self-neglecting; we write f SN. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 16 / 40

17 Class BSV selected properties If f BSV is continuous, then f SN. f SN if and only if it has the representation t f (t) = ϕ(t) ψ(s) ds, 0 where lim t ϕ(t) = 1 and ψ is continuous with lim t ψ(t) = 0. If f BSV is continuous, then there exists g C 1 such that f (t) g(t) and g (t) 0 as t.... Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 17 / 40

18 Regular variation and DEs Theory of RV is non-trivially used; assumptions, statements, proofs. Remark: In dozens of works on DEs, their authors work with regular variation without realizing it. Linear and half-linear DEs Geluk, Grimm, Hall, Jaroš, Kusano, Marić, Omey, Ř., Taddei, Tanigawa, Tomić Quasilinear DEs and systems (equations of Emden-Fowler, Lane-Emden, Thomas-Fermi types) Avakumović, Evtukhov, Jaroš, Kharkov, Kusano, Manojlović, Marić, Matucci, Radasin, Ř., Taliaferro, Tanigawa PDEs Cîrstea, Rădulescu, Zhang (uniqueness and asympt. behavior for solutions of nonlinear elliptic problems with boundary blow up); Niethammer, Pego (Ostwald ripening); Taliaferro (almost radial symmetry) Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 18 / 40

19 Regular variation and DEs Stochastic (P)DEs and Volterra equations Appleby, Debussche, Högele, Imkeller, Patterson Difference equations Ganelius, Geronimo, Manojlović, Kharkov, Kooman, Matucci, Ř., Smith, Van Assche (incl. orthogonal polynomials) q-difference equations and dynamic equations on time scales (dependence on the graininess) Ř., Vítovec... Saari (n-body problem), Mijajlović, Segan (Friedmann equations),... Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 19 / 40

20 Emden-Fowler type system Consider the quasilinear (or the generalized Emden-Fowler) equation where α, β > 0, p(t) > 0. (Φ α (y )) = p(t)φ β (y), Φ λ (u) = u λ sgn u, (E) Basic classification of nonoscillatory solutions (with respect to the limit behavior as t of a solution and its quasiderivative). Problems of (non)existence in the classes. A more precise description of asymptotic behavior of solutions (along with asymptotic formulae). In particular, we are interested in somehow difficult classes, e.g., the strongly increasing solutions (SIS), i.e., lim t y(t) = = lim t y (t). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 20 / 40

21 (Φ α (y )) = p(t)φ β (y), Φ λ (u) = u λ sgn u, (E) Kamo and Usami (2000, 2001) assumed p(t) t σ and α > β > 0. They established conditions (in terms of α, β, σ) guaranteeing that SIS solutions y of (E) have the form y(t) Kt γ, where K = K (α, β, σ), γ = γ(α, β, σ). The key role was played by the asymptotic equivalence theorem which says, roughly speaking: If the coefficients of two equations of the form (E) are asymptotically equivalent, then their solutions which are of the same class are also asymptotically equivalent. An extension of the classical results by Bellman, who assumed p(t) = t σ. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 21 / 40

22 (Φ α (y )) = p(t)φ β (y), Φ λ (u) = u λ sgn u, p(t) t σ We are interested in a generalization A regularly varying coefficient A general coefficient in the differential term Regularly varying nonlinearities Coupled systems Second order systems of k equations (even-order scalar equations) First order systems of n equations (n may be even or odd) An asymptotic equivalence theorem cannot be used. The theory of RV in combination with some other tools finds the application. Much wider class of equations can be included. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 22 / 40

23 Nonlinear system (of Emden-Fowler type) x 1 = a 1 (t)f 1 (x 2 ), x 2 = a 2 (t)f 2 (x 3 ),. x n 1 = a n 1 (t)f n 1 (x n ), x n = a n (t)f n (x 1 ), n N, n 2. a i are continuous, eventually of one sign, and (S) a i RV(σ i ), σ i R, i = 1,..., n, F i are continuous with uf i (u) > 0 for u 0, and F i ( ) RV(α i ), resp. F i ( ) RV 0 (α i ), α i (0, ), i = 1,..., n. A modification of our approach works also for some (more general) perturbed systems. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 23 / 40

24 Special cases n-th order two term nonlinear DE s x (n) = p(t)φ β (x) or, more general, D qn (γ n )D qn 1 (γ n 1 ) D q1 (γ 1 )x(t) = p(t)φ β (x), where D qi (γ i )x(t) = d dt (q i(t)φ γi (x)) The order can be even as well as odd. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 24 / 40

25 Special cases equations with a generalized Laplacian and/or an RV nonlinearity on the RHS For example, the second order equation with (r(t)g(x )) = p(t)f(x), - a generalized Laplacian (may include the classical p-laplacian operator or the curvature operator or the relativity operator or...) - a regularly varying nonlinearity on the right-hand side Typical examples of nonlinearities (we do not require monotonicity): F i (u) = Φ α (u) = u α sgn u (for second order equations or systems it may lead to the classical Laplacian operator). F i (u) = Φ α (u)l(u), with α > 0, where L(u) c (0, ), or L(u) = ln u γ 1 ln ln u γ 2. F i (u) = u α (A + Bu β ) γ ; a special choice yields u or u. 1 + u 2 1 u 2 Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 25 / 40

26 Partial differential systems System (S) includes also second order nonlinear systems of the form (A 1 (t)φ λ1 (y 1 )) = B 1 (t)g 1 (y 2 ), (A 2 (t)φ λ2 (y 2 )) = B 2 (t)g 2 (y 3 ),. (A k (t)φ λk (y k )) = B k (t)g k (y 1 ), which play important role in the study of positive radial solutions to the partial differential system div( u 1 λ1 1 u 1 ) = ϕ 1 ( z )G 1 (u 2 ), div( u 2 λ2 1 u 2 ) = ϕ 2 ( z )G 2 (u 3 ), Systems of Lane-Emden type.. div( u k λ k 1 u k ) = ϕ k ( z )G k (u 1 ). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 26 / 40

27 Extremal solutions x n+1 means x 1. x i = a i (t)f i (x i+1 ), i = 1,..., n, Conditions on coefficients: a i RV(σ i ), i = 1,..., n. Conditions on nonlinearities: F i RV(α i ); nonlinearities do not need to be monotone. Subhomogeneity : α 1 α n < 1 Let us examine, for example, the class (S) SIS = { (x 1,..., x n ) IS : lim t x i (t) =, i = 1,..., n } ; the so-called strongly increasing solutions (or fast growing solutions). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 27 / 40

28 The constants ν i and h i and SV functions L i depend on the coefficients and the nonlinearities; they can be explicitly computed. Theorem If ν i > 0, i = 1,..., n, then there exists (x 1,..., x n ) SIS (RV(ν 1 ) RV(ν n )) and (for every such a solution) x i (t) h i t ν i L i (t) as t, i=1,...,n. (AF) If ν i > 0 and, in addition, F i = Φ αi, i = 1,..., n, then SIS and for EVERY (x 1,..., x n ) SIS, one has (x 1,..., x n ) RV(ν 1 ) RV(ν n ) and (AF) holds with L F1 L Fn 1, L Fi (t) = t α i Fi (t), i = 1,..., n. ν i = ν i (σ 1,..., σ n, α 1,..., α n ), h i = h i (σ 1,..., σ n, α 1,..., α n ), L i = L i (σ 1,..., σ n, α 1,..., α n, L a1,..., L an, L F1,..., L Fn ). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 28 / 40

29 Sketch of the proof (the first part) Properties of RV functions are frequently used... The Schauder-Tychonoff fixed point theorem: We obtain a solution (x 1,..., x n ) SIS such that c i t ν i Li (t) x i (t) d i t ν i Li (t) for some constants c i, d i, i = 1,..., n. lim inf t x i (λt)/x i (t), lim sup t x i (λt)/x i (t) (0, ). The uniform convergence theorem and some other tools yield that lim t x i (λt)/x i (t) exists; in fact, Thus, RV follows. lim sup x i (λt)/x i (t) λ ν i lim inf x i(λt)/x i (t). t t Playing with asymptotic relations and the Karamata theorem yield the asymptotic formula. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 29 / 40

30 Sketch of the proof (the second part) SIS follows from the previous part. We take an arbitrary (x 1,..., x n ) SIS. Several auxiliary (quite) technical results, the generalized AM-GM inequality and some other estimates, and the properties of RV functions yield c i t ν i Li (t) x i (t) d i t ν i Li (t), i = 1,..., n. x i RV(ν i ) and the asymptotic formula follow by similar arguments as in the first part. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 30 / 40

31 We get back the Kamo-Usami result as a very special case. In other ( less special ) cases the results are new; even when a i (t) t σ i (for higher order) or for second order equations (with RV coefficients). For example, let p(t) = t ϱ L p (t) with L p SV and 0 < β < 1. If ϱ β(n 1) > 0, then the equation x (n) = p(t)φ β (x) possesses a solution x such that lim t x (i) (t) =, i = 0,..., n 1, and for any such a solution there hold ( ) ϱ + n n x RV and x 1 β (t) t ϱ+n 1 β L p (t) 1 β ϱ + n (1 β)(j 1). j=1 Regular variation of all these solutions is normalized. We can treat also some equations with non-rv coefficients via a suitable transformation. Other types of solutions (decreasing,...). Other types of equations (perturbed equations, singular equations, superlinearity,...) Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 31 / 40

32 Half-linear differential equations (r(t)φ(y )) = p(t)φ(y), where r, p C([a, )), r(t) > 0, Φ(u) = u α 1 sgn u with α > 1. (HL) The solution space is homogeneous but not additive. Bihari (1957, 1964), Elbert (1979), Mirzov (1976),... dozens of authors... O. Došlý, P. Ř., Half-Linear Differential Equations, Elsevier, North Holland, If α = 2, then (HL) reduces to a linear Sturm-Liouville equation. Radially symmetric solutions of a certain PDE with p-laplacian. If the dimension N = 1, then (PDE) (HL). Special (bordeline) case of quasilinear (generalized Emden Fowler) DEs. α-degree functionals (variational principle, Hardy type inequalities,...) Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 32 / 40

33 Solutions in the class Γ (Φ(y )) = p(t)φ(y), where p C([a, )), p(t) > 0, Φ(u) = u α 1 sgn u with α > 1. (HL) Theorem If p 1 α BSV, then all solutions of (HL) belongs to Γ(v) or Γ (v), where ( ) 1 α 1 α v =. p Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 33 / 40

34 Sketch of the proof (here only for increasing solutions; decreasing solutions are more difficult) First assume p C 1. Then (p 1 α ) (t) 0 as t. Take y such that y > 0, y > 0. Set w = p 1 β Φ(y /y). w satisfies the generalized Riccati equation ( ) w p (t) = 1 (α 1)w p 1 α (t) αp α+1 α (t) + w β 1 (β is the conjugate number to α). w satisfies lim t w(t) = (α 1) 1 β. y satisfies y Γ ( ( α 1 p ) 1 ) α. y (t)y(t) y 2 (t) 1 Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 34 / 40

35 Sketch of the proof (continuation) We drop the assumption on differentiability of p. Since p BSV there exists ˆp C 1 with ˆp(t) p(t) and (ˆp 1 α ) (t) 0 as t. For ε (0, 1) we consider the auxiliary equations (Φ(u )) = (1 + ε)ˆp(t)φ(u) and (Φ(v )) = (1 ε)ˆp(t)φ(v). For increasing solutions u, v we show ( u ) (t) α 1 ˆp 1 β (t) u(t) ( ) ε α 1 β, ( v (t) v(t) From the theorem on differential inequalities, lim t ( y (t) y(t) ) α 1 p 1 β (t) = (α 1) 1 β. ) α 1 ˆp 1 β (t) The rest of the proof is the same as in the previous part. ( ) 1 1 ε β α 1 Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 35 / 40

36 Remarks A part of the statement is a half-linear extension of known result. The other is new also in the linear case. (The Wronskian identity, reduction of order are not at disposal,...) By-product: Condition for rapid variation of all solutions (open problem solved). By-product: There are solutions y 1, y 2 such that ( ) 1 p(t) y i (t) ± α yi (t). α 1 Half-linear extension of Hartman s-wintner s result. Another generalization can be made. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 36 / 40

37 Complete classification of solutions (r(t)φ(y )) = p(t)φ(y), Karamata theory, de Haan theory, generalized Riccati technique, reciprocity principle, asymptotic linearization, and other tools are used to obtain complete classification of solutions in the framework of regular variation. Relations among the classes; description of the structure of the solution space. Asymptotic formulae for ALL solutions. Interesting phenomena (that are not known from the linear case) are revealed Quite general setting that enables us to cover many important cases. SV components of the coefficients play an important role; more complex structure than with asymptotically-power coefficients. Sufficiency and necessity of the conditions. New also in the linear case. An example of the result: (HL) Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 37 / 40

38 Consider the equation (Φ(y )) = p(t)φ(y). DS: eventually positive decreasing solutions; DS A,B : DS solutions with y A, y B; S (N )SV : (N )SV-solutions. Theorem Let p RV( α). If L p (t) 0 as t, where L p is SV component of p, then DS N SV. For y DS, one has y Π( ty (t)). Moreover, (i) If (sp(s)) 1 α 1 ds =, then a y(t) = exp { t as t, and S SV = S N SV = DS = DS 0,0. (ii) If (sp(s)) 1 α 1 ds <, then a y(t) = y( ) exp a { t ( ) 1 } sp(s) α 1 (1 + o(1)) ds α 1 ( ) 1 } sp(s) α 1 (1 + o(1)) ds α 1 as t, and S SV = S N SV = DS = DS B,0, B > 0. In addition, y( ) y(t) SV, L β 1 p (t)/(y( ) y(t)) = o(1). Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 38 / 40

39 Sketch of the proof Take y DS. We show that ty (t)/y(t) 0 as t, which implies y N SV. From Φ(y ) RV(1 α), we get y(t) Π( ty (t)) Set h(t) = t α 1 Φ(y (t)) (α 1) t a sα 2 Φ(y (s)) ds. Then h Π( (α 1)t α 1 Φ(y (t))) and h Π(th (t)). From the uniqueness of the auxiliary function up to asymptotic equivalence and h (t) = t α 1 p(t)φ(y(t)), we get ( y (t) y(t) = (1 + o(1)) p(t), where p(t) = tp(t) α 1 ) 1 α 1 as t. We distinguish the cases p(s) ds = and p(s) ds <. Then we integrate the asymptotic relation to obtain formulas. Moreover, we find y(t) 0 resp. y(t) y( ) (0, ). Properties of RV and Π functions yield the rest. Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 39 / 40

40 S. Matucci, P. Řehák, Extremal solutions to a system of n nonlinear differential equations and regularly varying functions, Mathematische Nachrichten 288 (2015), P. Řehák, On certain asymptotic class of solutions to second order linear q-difference equations, J. Phys. A: Math. Theor. 45 (2012) P. Řehák, Nonlinear Differential Equations in the Framework of Regular Variation, AMathNet 2014, 207 pp. P. Řehák, Asymptotic formulae for solutions of half-linear differential equations, Appl. Math. Comp. 292 (2017), P. Řehák, V. Taddei, Solutions of half-linear differential equations in the classes Gamma and Pi, Differ. Integral Equ. 29 (2016), P. Řehák, J. Vítovec, Regular variation on measure chains, Nonlinear Analysis 72 (2010), Pavel Řehák (IM CAS) RV and DEs External meeting, Brno 40 / 40